Rich Audu hit a pair of key 3-pointers in the second half for the Saints (4-13, 2-4), who had lost 11 in a row before defeating Canisius 57-54 Friday night.
Saint Peter's (6-10, 1-5) fell behind by 13 points in the first half but trailed just 49-47 with 5:07 remaining.
Siena ran off the next eight points, including consecutive treys by Audu, who played the rest of the way after Ryan Oliver injured his knee in the game's second minute.
The Saints had a huge edge in free throws, making 27 of 36, while Saint Peter's made 5 of 12. Anosike, who was averaging 45.8 percent from the line, made 12 of 17.
Desi Washington led the Peacocks with 16 points.